Finite Group Databasebeta

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#72236879

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#32947 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#11713271
commutator, cosocle#5950C23
FrattiniC22×C4C22×S4
Fitting#32947D3
hypercenterC8#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C4 ◁ C8

952 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,15,17,5,6,16,4,18)(2,27,21,22,8,26,10,11)(3,31,28,14,12,30,13,29)(7,32,23,24,9,25,19,20);(2,7)(3,12)(4,15)(8,9)(10,25)(11,20)(13,30)(14,29)(16,17)(19,26)(21,32)(22,24)(23,27)(28,31);(1,2,3,9,6,8,12,7)(4,10,13,23,17,21,28,19)(5,11,14,24,18,22,29,20)(15,26,31,32,16,27,30,25)

Polycyclic presentation:

2^4,3,2^5|a^2=d,d^2=j,f^2=j,g^2=j,b^a=bd,c^a=cj,d^b=dj,e^b=e^2,e^c=e^2,f^b=fgij,f^c=fg,f^e=g,g^b=gh,g^c=gj,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^b=hi,i^c=hi,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240981, #72240988, #72240996, …
maximal normal subgroup of#72240981, #72240988, #72240996, …
maximal quotient of#72341742, #72341764, C2×#72236879, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);